A304117 If n = Product (p_j^k_j) then a(n) = Product (pi(p_j)*k_j), where pi() = A000720.

1, 1, 2, 2, 3, 2, 4, 3, 4, 3, 5, 4, 6, 4, 6, 4, 7, 4, 8, 6, 8, 5, 9, 6, 6, 6, 6, 8, 10, 6, 11, 5, 10, 7, 12, 8, 12, 8, 12, 9, 13, 8, 14, 10, 12, 9, 15, 8, 8, 6, 14, 12, 16, 6, 15, 12, 16, 10, 17, 12, 18, 11, 16, 6, 18, 10, 19, 14, 18, 12, 20, 12, 21, 12, 12, 16, 20, 12, 22, 12

Offset: 1

Author

Ilya Gutkovskiy, May 06 2018

Keywords

• nonn
• mult

Examples

			a(36) = 8 because 36 = 2^2*3^2 = prime(1)^2*prime(2)^2 and 1*2*2*2 = 8.
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1000
• Ilya Gutkovskiy, Extended graphical example
• Index entries for sequences computed from indices in prime factorization
• Index entries for sequences computed from exponents in factorization of n

Crossrefs

Cf. A000026, A000040, A000720, A001414, A002110, A003963, A005361, A056239, A156061, A304037.

Programs

Mathematica

a[n_] := Times @@ (PrimePi[#[[1]]] #[[2]] & /@ FactorInteger[n]); a[1] = 1; Table[a[n], {n, 1, 80}]

PARI

a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = primepi(f[k,1])*f[k,2]; f[k, 2] = 1); factorback(f); \\ Michel Marcus, May 06 2018

Formula

a(n) = A005361(n)*A156061(n).

a(p^k) = A000720(p)*k where p is a prime.

a(A002110(m)^k) = k^m*m!.

As an example:

a(A000040(k)) = k.

a(A006450(k)) = A000040(k).

a(A001248(k)) = a(A031215(k)) = A005843(k).

a(A030078(k)) = a(A031336(k)) = A008585(k)

a(A061742(k)) = A000165(k).

a(A115964(k)) = A032031(k).

a(A002110(k)) = A000142(k).

a(A080696(k)) = A002110(k).